Night reader wrote:If u-v=a, u^2-v^2=b and u^3-v^3=c, find the equation of a, b and c
A. 3b+2a^2=2ac
B. b+4a^2=2c
C. 3b^2+a^4=4ac
D. 3b^2+a^2=4ac
E. b+4a^2=4c
Hi there!
Please note that this question stem should in fact be "which of the following equations relates a, b and c ?" , therefore we have to look at the alternatives to "focus" on what we should do!
(A) 3b+2a^2 has no terms in u^4 but 2ac has, therefore it is not alternative (A).
(B) b+4a^2 has no terms in u^3 but 2c has, therefore it is not alternative (B).
(C) Please wait till the end.
(D) Left side has a "u*v" term, right side has not, therefore it is not alternative (D).
(E) Left side has no u^3 term, right side has, therefore it is not alternative (E).
In the GMAT, I would go for (C) by exclusion, but now we are studying for the GMAT, so let´s check it:
3b^2 + a^4 = 3(u-v)^2*(u+v)^2 + (u-v)^4 =
= (u-v)^2 *[3(u+v)^2+(u-v)^2] = ... = (u-v)^2*4*(u^2+v^2+uv)
Now please remember that (u-v)(u^2+v^2+uv) = u^3-v^3 what motivates me to do:
(u-v)^2*4*(u^2+v^2+uv) = 4(u-v)(u-v)(u^2+v^2+uv) = 4(u-v)(u^3-v^3) = 4ac , and we proved (C) is really correct.
Best Regards,
Fabio.
